Presentations, Conjugacy, Roots, and Centralizers in Groups of Piecewise Linear Homeomorphisms of the Real Line

Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line

“…In any group, a classification of reversible maps follows from a classification of conjugacy. It is known how to classify conjugacy in H(R), PLF(R), and PL(R) (see [3,7]). We show in §2 that H + (R) and PL + (R) both contain non-trivial reversible maps whereas PLF + (R) contains none.…”

“…For example, the map x → max 2x, 1 2 x is reversed by x → −x. A complete classification of the reversible members of PLF(R) follows immediately from the conjugacy classification in [3].…”

“…One of these three quantities is Γ f , which we have already encountered. The reader should consult [3] for definitions of the other two quantities; they are not needed here.…”

“…In contrast, there are uncountably many conjugacy classes of fixed point free elements in PLF(R). (See [3] …”

“…In [3], Brin and Squier characterise the conjugacy classes in PLF + (R) using geometrical invariants. In particular, for any function f in PLF + (R), they describe three quantities, which we collectively denote by Σ f , with the property that maps f and g in PLF + (R) are conjugate if and only if Σ f = Σ g .…”